By the time an auctioneer shouts "Sold!" most
bidders have already gone too far

By Michael Abrams

DISCOVER Vol. 23 No. 08 | August 2002

A third of the way into A Beautiful
Mind, the recent film about mathematician John Nash, a
sexy blonde and four brunettes walk into a bar, batting their
eyelashes. After a bit of ogling, Nash and his number-minded
friends decide to compete for the blonde. Then Nash has second
thoughts. If everyone goes for the same woman, he says, we'll
just end up blocking each other out and offending the rest of
the women. The only way for everyone to succeed is to
ignore the blonde and go for the brunettes instead.

Illustration by
David Plunkert

The scene is an
attempt to illustrate Nash's most important contribution to
game theory—the Nash equilibrium. Nash showed that in any
competitive situation—war, chess, even picking up a date at a
bar—if the participants are rational, and they know that their
opponents are rational, there can be only one optimal
strategy. That theory won Nash a Nobel Prize in economics. The
math behind it is flawless and has transformed the way people
think about evolution, arms races, stock markets, and
ticktacktoe. There's only one problem: People aren't
rational.

"They just aren't that smart," says Thomas
Palfrey, an economist at the California Institute of
Technology, "and if they are that smart, they don't think
everybody else is that smart." Palfrey would fit right in at a
greyhound track, with his swept-back hair and
confidence-winning eyes. He says "basically" and "actually" a
lot and discusses his work as if letting you in on a hot tip
for the second race. The best strategy in any game against
Palfrey is not to play at all: In the past few years, he has
hit upon a variation on Nash's theory that describes true
competitive behavior for the first time. More than any
economist before him, Palfrey is likely to guess your every
move.

The imbalance at the crux of the Nash equilibrium
first struck him and his colleague Richard McKelvey in 1988,
at a conference in Haifa, Israel. Game theorist Robert Aumann
was giving a talk on a hypothetical game called centipede, in
which two players take turns being offered the larger portion
of a pot of money. If either player accepts, the game is over,
and the loser gets the smaller amount. But each time the pot
is declined, the amount of money inside grows. If it increases
tenfold, a nervy player's winnings can grow from $10 to $10
million in only six turns. But the Nash equilibrium dictates a
less daring strategy—take the larger pot the first chance you
get. "This is ridiculous," Palfrey remembers thinking. "Game
theorists keep talking about the centipede game as if they
know what's going to happen, but in fact they don't know. So
let's run an experiment."

Back at Caltech, he and
McKelvey sat some students in front of a network of cubicled
computers and let them play one another anonymously. "This was
the first time I'd ever run an experiment that was done just
to see what was going to happen," Palfrey says. "It was sort
of like when you got your chemistry set for Christmas in fifth
grade and you saw all these different chemicals and sort of
threw them all in together to see what would happen. But we
got really amazing data." Contrary to the predictions of the
Nash equilibrium, only 37 of 662 centipede games ended in the
first round. "The problem we had then was how to explain the
data. How would you actually model players not being fully
rational?"

The breakthrough came when the economists
factored in altruism and skepticism. The players knew that
their decisions didn't always make sense—and that their
opponents weren't always rational, either—and that made their
optimal strategy distinctly different from what Nash
predicted. "You can never know what somebody else is
thinking," Palfrey says. "All you can do is try to guess what
they're thinking. And they're trying to guess what you're
thinking, too." To model this behavior, Palfrey mixed a little
statistics in with the theory behind the Nash equilibrium.
Strategic mistakes and errors are part of the new formula, but
the reasoning is the same as Nash's. There's still a single
best strategy: Palfrey calls it the quantal response
equilibrium.

In 1998, not long after Palfrey and
McKelvey's ideas coalesced, Palfrey started running model
auctions to test the theory. For years in economic circles, a
controversy had raged over the subject of overbidding. Why is
it that people in auctions routinely pay more than they
should? Some economists, like Glenn Harrison of the University
of South Carolina, thought that bidders simply value the joy
of winning more than the satisfaction of getting an object for
less than it's worth—especially when it's not worth much to
begin with. Others, like Daniel Friedman at the University of
California at Santa Cruz, thought overbidding might be the
result of an aversion to risk: Most bidders would rather bid
too high than run the risk of losing the item
altogether.

To see who was right, Palfrey had a group
of students join a series of computer auctions using real
money. Each item up for auction was assigned a value. Then
pairs of students bid against each other, with each student
allowed a single bid. When a student won an item for less than
its assigned value, he kept the difference. For example, if
Palfrey told a student an object was worth $7, and the student
won it with a $3 bid, he would win $4. Each student
participated in 15 separate auctions. It was, Palfrey says, "a
very simplified version of the New York Stock
Exchange."

Every auction has a Nash equilibrium—a bid
that perfectly balances the risk of losing to a higher bidder
(and making no profit) against the possibility of greater
profits (the lower you bid, the more money you'll make). In
Palfrey's experiment, that optimal bid was half the object's
given value, yet the students regularly bid much higher.
According to the Nash equilibrium, they should have walked out
of the experiment with an average of $14.20. Instead, they
averaged only $10.70—exactly as Palfrey's quantal response
equilibrium had predicted.

The experiment banged the
gavel on the overbidding debate. Risk aversion and random
mistakes pushed the bids higher—even in auctions for
inexpensive objects. When overbidding was riskier than
underbidding, the students were less likely to overbid. To
Palfrey, bidders are like people in a parking lot faced with
feeding the meter or potentially paying a parking ticket:
They'll feed the meter even if the attendant only comes by
once a year. "The more risk averse you are in an auction, the
higher you bid," Palfrey says. And the more bidders overbid,
the more their high bids begin to snowball.

These days
Palfrey is testing quantal response on "all-pay auctions,"
where everyone who bids has to pay up—whether they win the
bidding or not. "It has applications to things you wouldn't
normally dream of," he says, like political campaigns, where
even losing candidates have to spend a fortune, and mating
contests in nature, where two males may fight to the death for
a single female. "The real power of this is that it's
replicable," Palfrey says. "It's not just your group of eight
people in a lab. Somebody could run the auction at Harvard,
and they'll see the same thing."

Even the most rational
people can be predictably irrational, Palfrey concludes. In
some business schools, professors make an exercise of
auctioning off a dollar to their students, stipulating that
both top bidders will have to pay. The best strategy is simply
not to bid, Palfrey says. "But then it's tricky, because if
everyone else realizes this, then why don't I bid 10 cents? It
would be irrational for anybody else to bid 11 cents. Well,
the problem is that some clown out there is going to bid 11
cents." Once the bidding gets going, he says, often only the
professor can stop it.